希尔密码_密码学中的希尔密码

悠悠 2023-03-05 09:47 42阅读 0赞

希尔密码

It is a polygraphic substitution cipher that depends on linear algebra. Every letter from the alphabet is represented by several modulo 26. Simply, we write as **A = 0, B = 1, ..., Z = 25** is used, but this is not a correct feature of the cipher.

这是一个依赖于线性代数的测谎替代密码。 字母表中的每个字母都由26为模。简单地，我们写为**A = 0，B = 1，...，Z = 25** ，但这不是密码的正确特征。

To encrypt a message, each block of **n** letters is multiplied by an invertible **n × n** matrix, against modulus 26 or we can say the multiple inverses of **n**.

为了加密消息，将**n个**字母的每个块都乘以可逆**n×n**矩阵(模数为26)或我们可以说**n**的多个逆。

To decrypt the message, every block is multiplied by the inverse of the matrix or inversed matrix used for encryption.

为了解密该消息，将每个块乘以用于加密的矩阵或逆矩阵的逆。

The matrix used for encryption is the plain text and one matrix is formed is cipher key when we combine or multiple them we get the new matrix called ciphertext, and the key should be chosen randomly from the set of invertible **n × n** matrices (modulo 26).

用于加密的矩阵是纯文本，当我们将它们组合或相乘时，形成的一个矩阵是密码密钥，得到一个新的矩阵，称为密文，密钥应从可逆**n×n**矩阵集中随机选择(取模26 )。

### 加密 **(**Encryption**)** ###

We have to do encryption on the message **'ACT' (n=3)**. The key is **'GYBNQKURP'** which can be written as the **n x n** matrix that is **3 x 3** matrix,

我们必须对消息**“ ACT”(n = 3)**进行加密。 关键是**“GYBNQKURP”，**其可以被写为**NxN**矩阵是**3×3**矩阵，

![Hill Cipher][]

The message for encryption **'ACT'** is written as a vector,

加密消息**“ ACT”**写为矢量，

![Hill Cipher 1][]

The enciphered vector is given as,

加密的向量为

![Hill Cipher 2][]

In this, we just do the multiple matrices we just multiple the **1** row with **1** column as we can say **6\*0+24\*2+1\*19=67** same as **13\*0+16\*2+10\*19=222** and **20\*0+17\*2+15\*19=319**.

在这里，我们只是做多个矩阵，我们只将**1**行与**1**列相乘，我们可以说**6 \* 0 + 24 \* 2 + 1 \* 19 = 67**与**13 \* 0 + 16 \* 2 + 10 \* 19 = 222相同**和**20 \* 0 + 17 \* 2 + 15 \* 19 = 319** 。

And that matrix does the **mod 26** with particular column and made again the **3\*1** matrix and we alphabetically write them with called ciphertext i.e **"POH"**.

然后，该矩阵使用特定的列进行**mod 26**运算，并再次制作**3 \* 1**矩阵，然后按字母顺序将其写为密文，即**“ POH”** 。

### 解密 **(**Decryption**)** ###

To decrypt the message, we turn the ciphertext back into a plain text, then simply multiply by the inverse matrix of the key matrix as **"IFKVIVVMI"** in letters. The inverse of the matrix used in the encryption is,

为了解密该消息，我们将密文转换回纯文本，然后简单地乘以密钥矩阵的逆矩阵，即字母**“ IFKVIVVMI”** 。 加密中使用的矩阵的逆是

![Hill Cipher 3][]

For ciphertext which we already find we decrypt that and make plain text,

对于我们已经发现的密文，我们将其解密并制成纯文本，

![Hill Cipher 4][]

Now, we just find the inverse matrix then multiple it by key then it will give us back **"ACT"**.

现在，我们只需找到逆矩阵，然后将其乘以键，然后它将返回**“ ACT”** 。

**Let, K is DDCF**

**设K为DDCF**

![Hill Cipher 5][]

be the key and suppose the plaintext is **"HELP"**. Then this plaintext is represented by two pairs because **n** is **2**.

是键，并假定明文为**“ HELP”** 。 然后，此纯文本用两对表示，因为**n**为**2** 。

![Hill Cipher 6][]

Firstly we do encryption, then we will compute

首先我们进行加密，然后我们将进行计算

![Hill Cipher 7][]

and continue encryption as follows and written as,

并继续如下加密并写为：

![Hill Cipher 8][]

The matrix **K** is invertible, hence **K\-1** exists such that **KK\-1 = K\-1K = I2** The inverse of **K** can be computed by using the formula,

矩阵**K**是可逆的，因此存在**K \-1** ，使得**KK \-1 = K \-1 K = I 2**可以使用公式计算**K**的逆，

![Hill Cipher 9][]

This formula still holds after a modular reduction if a modular multiplicative inverse is used to compute.

如果使用模块化乘法逆来进行计算，则该公式在模块化归约后仍然成立。

Now, we have to do decryption,

现在，我们必须进行解密，

![Hill Cipher 10][]

Then we compute,

然后我们计算

![Hill Cipher 11][]

And we get,

我们得到

![Hill Cipher 12][]

**Security**

**安全**

The basic **Hill Cipher** is vulnerable to a known-plaintext attack that attacks by key because it is completely linear algebra. An opposite site that intercepts n plaintext/ciphertext character pairs can set up a linear system that can be easily solved; if this will happen then this system is undefined, it is the only way is to add a few more plaintexts/ciphertext pairs. While when **n x n** matrix multiplication is a useful step when it will be combined with other non-linear operations because matrix multiplication can provide diffusion. For example, a unique chosen matrix can give security that minor differences before the matrix multiplication will give the answer in huge differences after the matrix multiplication. Otherwise, some new ciphers use a matrix multiplication step to gave diffusion. For example, the MixColumns matrix step in AES cipher is matrix multiplication. The function g in Twofish is a combination of non-linear algebra S-boxes i.e substitution boxes with a carefully chosen matrix multiplication (MDS) is used this.

基本的**希尔密码**很容易受到已知明文攻击，因为它是完全线性的代数，因此会受到密钥的攻击。 截获n个纯文本/密文字符对的相对站点可以建立一个易于解决的线性系统； 如果将发生这种情况，则该系统是不确定的，唯一的方法是添加更多的明文/密文对。 当**nxn**矩阵乘法是有用的步骤时，它将与其他非线性运算结合使用，因为矩阵乘法可以提供扩散。 例如，唯一选择的矩阵可以确保矩阵乘法之前的微小差异将在矩阵乘法之后的巨大差异中给出答案。 否则，某些新密码将使用矩阵乘法步骤进行扩散。 例如，AES密码中的MixColumns矩阵步骤是矩阵乘法。 Twofish中的函数g是非线性代数S-box的组合，即，使用具有精心选择的矩阵乘法(MDS)的替换框。

**Reference:** [Hill Cipher][Hill Cipher 13]

**参考：** [希尔密码][Hill Cipher 13]

> 翻译自: [https://www.includehelp.com/cryptography/hill-cipher.aspx][https_www.includehelp.com_cryptography_hill-cipher.aspx]

希尔密码

[Hill Cipher]: /images/20230208/e4bb62460efa48bd98d4ea1657418f62.png
[Hill Cipher 1]: /images/20230208/d26e24aeb51642ea8c9e2d7c9a3cd819.png
[Hill Cipher 2]: https://www.includehelp.com/cryptography/Images/hill-cipher-2.jpg
[Hill Cipher 3]: https://www.includehelp.com/cryptography/Images/hill-cipher-3.jpg
[Hill Cipher 4]: https://www.includehelp.com/cryptography/Images/hill-cipher-4.jpg
[Hill Cipher 5]: /images/20230208/5b4bbaeacf424f11b7ce350710034580.png
[Hill Cipher 6]: /images/20230208/5875923f7b8e451fa02e7d09e4626911.png
[Hill Cipher 7]: /images/20230208/092273545909483e83705f2b95d39a55.png
[Hill Cipher 8]: /images/20230208/70e90da895bc4438821f2fea248e8aaa.png
[Hill Cipher 9]: /images/20230208/c11b602cca93433f8de78ce263d3af7a.png
[Hill Cipher 10]: /images/20230208/9224ca92587546e7856a6ff59de29ba9.png
[Hill Cipher 11]: https://www.includehelp.com/cryptography/Images/hill-cipher-11.jpg
[Hill Cipher 12]: /images/20230208/9aafd952da36478b902b14e6bb49a5f9.png
[Hill Cipher 13]: https://en.wikipedia.org/wiki/Hill_cipher
[https_www.includehelp.com_cryptography_hill-cipher.aspx]: https://www.includehelp.com/cryptography/hill-cipher.aspx